A Hölder estimate for non-uniform elliptic equations in a random medium

Uniform regularity for second order elliptic equations in a highly heterogeneous random medium is concerned. The medium is separated by a random ensemble of simply closed interfaces into a connected sub-region with high conductivity and a disconnected subset with low conductivity. The elliptic equations, whose diffusion coefficients depend on the conductivity, have fast diffusion in the connected sub-region and slow diffusion in the disconnected subset. Without a stationary–ergodic assumption, a uniform Hölder estimate in ω,ϵ,λ for the elliptic solutions is derived, where ω is a realization of the random ensemble, ϵ∈(0,1] is the length scale of the interfaces, and λ²∈(0,1] is the conductivity ratio of the disconnected subset to the connected sub-region. Results show that if external sources are small enough in the disconnected subset, the uniform Hölder estimate in ω,ϵ,λ holds in the whole domain. If not, it holds only in the connected sub-region. Meanwhile, the elliptic solutions change rapidly in the disconnected subset.